Integrand size = 19, antiderivative size = 87 \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {1}{2 a x^2}+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x^2}+\frac {b^2 \left (c x^n\right )^{2/n} \log (x)}{a^3 x^2}-\frac {b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x^2} \]
-1/2/a/x^2+b*(c*x^n)^(1/n)/a^2/x^2+b^2*(c*x^n)^(2/n)*ln(x)/a^3/x^2-b^2*(c* x^n)^(2/n)*ln(a+b*(c*x^n)^(1/n))/a^3/x^2
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {a^2-2 a b \left (c x^n\right )^{\frac {1}{n}}-2 b^2 \left (c x^n\right )^{2/n} \log (x)+2 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{2 a^3 x^2} \]
-1/2*(a^2 - 2*a*b*(c*x^n)^n^(-1) - 2*b^2*(c*x^n)^(2/n)*Log[x] + 2*b^2*(c*x ^n)^(2/n)*Log[a + b*(c*x^n)^n^(-1)])/(a^3*x^2)
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {892, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-3/n}}{b \left (c x^n\right )^{\frac {1}{n}}+a}d\left (c x^n\right )^{\frac {1}{n}}}{x^2}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \left (\frac {\left (c x^n\right )^{-3/n}}{a}-\frac {b \left (c x^n\right )^{-2/n}}{a^2}+\frac {b^2 \left (c x^n\right )^{-1/n}}{a^3}-\frac {b^3}{a^3 \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )}\right )d\left (c x^n\right )^{\frac {1}{n}}}{x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {b^2 \log \left (\left (c x^n\right )^{\frac {1}{n}}\right )}{a^3}-\frac {b^2 \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3}+\frac {b \left (c x^n\right )^{-1/n}}{a^2}-\frac {\left (c x^n\right )^{-2/n}}{2 a}\right )}{x^2}\) |
((c*x^n)^(2/n)*(-1/2*1/(a*(c*x^n)^(2/n)) + b/(a^2*(c*x^n)^n^(-1)) + (b^2*L og[(c*x^n)^n^(-1)])/a^3 - (b^2*Log[a + b*(c*x^n)^n^(-1)])/a^3))/x^2
3.31.12.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1)) Subst[Int[x^m*(a + b *x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x ] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.35 (sec) , antiderivative size = 302, normalized size of antiderivative = 3.47
method | result | size |
risch | \(-\frac {\left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}} b^{2} \ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}{a^{3} x^{2}}+\frac {b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{a^{2} x^{2}}-\frac {1}{2 a \,x^{2}}+\frac {c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}} b^{2} \ln \left (x \right )}{a^{3} x^{2}}\) | \(302\) |
-1/a^3*((x^n)^(1/n))^2*(c^(1/n))^2/x^2*exp(I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n )+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)*b^2*ln(b*(x^n)^(1/n)*c^(1/n) *exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I *c*x^n))/n)+a)+1/a^2*b/x^2*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)* (-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)-1/2/a/x^2+1/a^3/ x^2*c^(2/n)*(x^n)^(2/n)*exp(I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n) )*(csgn(I*c)-csgn(I*c*x^n))/n)*b^2*ln(x)
Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) - 2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \left (x\right ) - 2 \, a b c^{\left (\frac {1}{n}\right )} x + a^{2}}{2 \, a^{3} x^{2}} \]
-1/2*(2*b^2*c^(2/n)*x^2*log(b*c^(1/n)*x + a) - 2*b^2*c^(2/n)*x^2*log(x) - 2*a*b*c^(1/n)*x + a^2)/(a^3*x^2)
\[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int \frac {1}{x^{3} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{3}} \,d x } \]
\[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int \frac {1}{x^3\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )} \,d x \]